On Wedderburn's division algebra theorem of 1914 (Q1276298)

Let \(L/K\) be a cyclic Galois extension of order \(n\), \(n>1\), let \(\theta\) be a generator of \(\text{Gal}(L/K)\). Let \(A=(L/K,\theta,a)\), \(a\in K\setminus\{0\}\), denote the cyclic algebra with basis as left \(L\)-vectorspace given by \(1,j,j^2,\ldots,j^{n-1}\) and as relations defining the multiplication in \(A\): for all \(\alpha\in L\): \(j\alpha=\theta(\alpha)\), \(j^n=a\). It is well known that \(A\) is a central simple \(K\)-algebra and that \(A\) is a division algebra if and only if \(a\) is a norm of \(L\), i.e. \(a=N_K^L(\alpha)\) for some \(\alpha\in L\). This fact is usually obtained as a corollary of more general results on crossed products and cohomology. In his 1914 paper ``A type of primitive algebra'' [Trans. Am. Math. Soc. 15, 162-166 (1914; JFM 45.0189.04)], \textit{J. H. M. Wedderburn} gave a (slightly weaker) sufficient condition for \(A\) to be a division algebra. He proved that if no power of \(a\) strictly less than the \(n\)-th power is a norm of \(L\) then \(A\) is a division algebra. In the paper McConnell gives a simple and self-contained proof of this fact. The author assumes \(A\) is not a division algebra and chooses a zero-divisor \(f\in A\) of the form \(f=j^r+\alpha_{r-1}j^{r-1}+\cdots+\alpha_0\) with \(1\leq r\leq n-1\), \(r\) being minimal such that \(f\) is a zero-divisor. He then shows, using a lemma describing multiplication of elements in \(A\), that \(a^r\) is a norm. The author announces the publication of a more detailed version of the paper, including more historical details.